INVESTMENT, TOBIN'S Q, AND. MULTIPLE CAPITAL INPUTS. Robert S. Chirinko. Working Paper No. 2033. NATIONAL BUREAU OF ECONOMIC RESEARCH.
NBER WORKING PAPER SERIES
INVESTMENT, TOBIN'S Q, AND MULTIPLE CAPITAL INPUTS
Robert S. Chirinko
Working Paper No. 2033
NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 October 1986
This paper is a revised version of "On The Inappropriateness Of An Empirical Q Investment Equation," and has benefited from the comments and advice offered by D. Cox, R. Ehrenberg, R. Eisner, R. Gordon, C. Hakkio, F. Hayashi, N. Jenkinson, N. Kiefer, 1. Mroz, L. Muus, G. Silverstein, K. Troske, L. Vincent, and participants at numerous seminars and the NBER Summer Institute. Financial support from the Social Science Research Council, the Hoover Institution, and the National Science Foundation under Grant No. SES—8309073 is gratefully acknowledged. Remaining errors, omissions, and conclusions are the sole responsibility of the author. The research reported here is part of the NBER's research program in Economic Fluctuations. Any opinions expressed are those of the author and not those of the National Bureau of Economic Research.
NBER Working Paper #2033 October 1986
Investment, Tobin's Q, and Multiple Capital Inputs
ABSTRACT
Despite
their solid theoretical basis, models of business
investment based on Tobin's Q
theory
have recorded a generally
This paper examines one possible source of misspecification. When the firm's technology is expanded to include two or more capital inputs, the investment disappointing empirical performance.
equation following from maximizing behavior includes Q as well as a series of additional explanatory variables. The importance of these
omitted variables is assessed, and the econometric evidence is mixed, as the Multi-Capital Q
model
clearly dominates the Conventional
In addition, the implications of the parameter estimates from the Conventional and specification but empirical problems remain.
Multi-Capital models for tax policy are noted.
Robert
S. Chirinko Committee on Policy Studies 1050 East 59th Street University of Chicago Chicago, IL 60637
INVESTMENT. TOBIN'S Q. AND MULTIPLE CAPITAL INPUTS
I. INTRODUCTION
Quantifying the parameters characterizing the capital accumulation process has been the subject of economic research for a
number of years. A substantial amount of our knowledge of this process is based on estimates of the "Neoclassical" investment model pioneered by Dale Jorgenson (1963). While this econometric specification is capable of explaining a substantial amount of the
variation in investment expenditures, it has been criticized for a lack of careful consideration of dynamics arising from expectations (Lucas, 1976) and intertemporal aspects of the technology (Nerlove, 1972).
The Q theory approach to investment behavior, introduced by Keynes (1964) and revitalized by James Tobin (1969), offers possible solutions to these two shortcomings. In this model, a forwardlooking firm faced with costs in adjusting its capital stock will have
its investment expenditures determined by marginal Q, the ratio of the discounted future revenues from an additional unit of capital to the net-of-tax purchase price. Whenever this ratio differs from unity, the firm has an incentive to alter its capital stock, but its actions are tempered by the adjustment cost technology.
Since
marginal Q is unobservable, empirical models have utilized average Q, defined as the ratio of the value of the firm, as evaluated in financial markets, to the replacement cost of its existing capital stock.
By relying on financial market data, which in principle incorporates expectations of future variables relevant to the investment decision,
2
Q models
provide a direct role for expectations in the econometric
specification.
The problem of unobservable expectations has received a great
deal of attention in the applied econometrics literature, and an alternative approach uses time-invariant, forecasting equations to calculate marginal Q. This class of solutions includes the two-step procedure of Abel and Blanchard (1984), the maximum likelihood
estimator of Hansen and Sargent (1980), and the Euler equation technique of Hansen and Singleton (1982); in the latter case, the forecasting equations are determined by the choice of instruments. These methods arose in response to the critique of Lucas (1976), who
argued that "any change in policy will systematically alter the structure of econometric models" (p. 41). Under this view, changes in policy are identified as changes in the parameters governing policy outcomes, and these parameters will generally affect forecasts of economic variables. Only if one maintains that the sample period contains no changes in policy or non-policy factors affecting the
stochastic environment will the forecasting solutions to the unobservable expectations problem be valid.
A significant advantage is that estimation can proceed even if the sample
of the Q approach period is characterized by an unstable stochastic environoment.
However, the usefulness of Q theory is called into question by its generally disappointing empirical performance.1 A number of the problems associated with empirical Q models are evident in the
1 See Chirinko (1986, Section V.B.3) for a review of the econometric results from Q models, and Clark (1979) for simulations comparing Q to other investment models.
3
following investment equation estimated with aggregate data for
equipment, structures, and inventories, 1(t) / K(t) =
.072 + .013
(.021)
= .256
(t) / K(t) +
(.005)
m = 3.477
.368
I(t-1) / K(t)
(.190)
(1)
Res Sum Sq = .5020
Sample Period: 1950-1978
where 1(t) and K(t) are the investment flow and capital stock, respectively, for the three capital goods and c (t) is a variable closely related to average Q (to be discussed below). The coefficient of .013 is smaller than that obtained in the Q study of Summers (1981, p. 101), who found that investment responded unreasonably slowly to variations in (t). His simulations indicated that, twenty years after
an unexpected change in the economic environment, the capital stock would have moved only three-fourths of the way to its ultimate, steady-state value.2 Contrary to the theory, the lagged dependent
2 When constrained by a geometric lag distribution, the econometric estimates of Ciccolo (1975) imply that the mean lag of the adjustment to a change in the long-run capital stock is seven years (Hall, 1977, p. 89).
4
variable proves an important determinant of investment.3 Furthermore, the R2 is rather low and, as measured by the mstatistic (cf., fn. 11), serial correlation in the residuals indicates that the model is misspecified. A problem thus facing the applied econometrician is that conventional Q theory does not suggest what modifications or additional explanatory variables may be useful in attenuating these deficiencies.
An examination of the investment data for equipment, structures, and inventories suggests a possible cause of this
In Table I, the serial correlation patterns for the three capital inputs differ widely, disappointing empirical performance.
indicating that the capital homogeneity assumption implicit in the conventional Q model may be inappropriate. Under the assumption
that adjustment costs increase with the durability of the capital asset, these patterns are consistent with the adjustment cost technology underlying the Q model. Thus, the central idea to be exploited in this study is that conventional formulations of empirical Q models
are misspecified by failing to recognize the possibility that the value of the firm depends on two or more capital inputs with differing adjustment cost technologies.
That the relationship
(1983) has shown that lagged Q should be an important determinant of investment spending. There are two objections to his result. First, he enters lags into the maximization problem by assuming that adjustment costs depend on both current and lagged investment, but it is not at all clear what phenomena are being captured by this formulation. Second, the investment schedule that follows from his model is quite different from those actually used in econometric work, and thus the relation betweeen his result and empirically significant lagged variables is not apparent. 3
Fischer
5
between Q
and
investment is sensitive to the number of capital
inputs has been noted previously (Chirinko, 1982b; Wildasin, 1984). In this paper, we investigate the specification error due to these
omitted variables, show that a structural model can be preserved under a broader specification of the technology, and provide econometric evidence to evaluate the Multi-Capital Q model. The theoretical development of the Multi-Capital Q model
is
contained in Section II, which relates investment expenditures, (unobserved) marginal Q,
and
(observed) average Q.
We give
particular attention to the way in which debt finance affects the definition of average Q.
Section
relating to the investment model. and Multi-Capital Q
models
econometric techniques.
III considers specification issues
In Section IV, the Conventional
are estimated with a variety of
The simulated responses of investment
expenditures and capital accumulation to an unanticipated change in the economic environment are calculated for both models in Section V. A summary and conclusions are provided in Section VI.
6
II. THE MULTI-CAPITAL Q MODEL - THEORY
The representative firm chooses variable and quasi-fixed inputs to maximize the present discounted value of its cash flow (V(O)) over an infinite horizon, 00
V(O) =
fo exp(- f p(s) ds) {R(t) -
C(t)
-
T(t) - F(t)} dt ,
(2)
0
where p (s) is the discounting factor at time s and R(t), C(t), T(t), and
F(t) represent revenues, operating costs, taxes, and net financing costs, respectively, at time t. Revenues (R(t)) are generated by selling a single product in a competitive market at an exogenous price (p(t)). The production technology (cI[tJ) depends on labor (L(t)) and J distinctive types of capital (K(t), jEJ), and is characterized by
constant returns to scale with respect to all inputs and by Inada conditions,
R(t) = p(t)
C1
[L(t),K(t),VjcJI
.
(3)
Operating costs (C(t)) arise from purchasing inputs and adjusting the capital stocks. The firm purchases labor services at
price w(t) and new capital (I(t)) at prices v(t) in perfectly competitive factor markets. To capture the quasi-fixed nature of capital, we assume that the firm incurs adjustment costs when incorporating new capital goods into the production process. These internal costs can be viewed as the movement of real resources from
producing output toward installing capital goods, are valued in terms
7
of the opportunity cost of foregone output, and increase at an increasing rate as investment exceeds replacement needs, defined as the product of an exponential depreciation rate () and the existing These properties apply to the adjustment cost functions (F[I(t),K(t)],VjcJ) that are homogeneous of degree one in capital stock.
both arguments,
w(t) L(t) +
C(t)
v(t) I(t) + jeJ
F[I(t),K(t)J ,
p(t)
(4)
jEJ
An income tax is assessed at rate t(t) against the firms revenues less labor and adjustment costs, and is reduced by tax credits (k3(t)) extended on the purchase of investment goods. The effective cost of the jth capital good is further lowered by tax depreciation allowances granted against current taxes but based on
both current and past investments, t
t(t) f D(t-s,s) va(s) Ii(s) ds
(5)
-00
where
D(t-s,s) are tax depreciation allowances per dollar of the jth
investment made t-s periods ago according to the tax code in effect in period s. When embedded in the firm's discounted cash flow expression (2), equation (5) can be rearranged to isolate those factors
that depend on current decisions and those that are predetermined at time t,
8
(6a)
v(t) (k(t)+z(t)) I(t) + A(t,O) , 00
z(t) =
ft
5
exp(- f p(u) du) er(s) D3(s-t,t) ds , t
(6b)
0
A(t,O) = t(t) f D(t-s,s) vs(s) Ia(s) ds .
(6c)
-00
te [0,00)
The expression for z(t) represents the present discounted value of current and future tax depreciation allowances flowing from a dollar
of investment in period t, and (6c) represents the total value of tax depreciation allowances claimed at time t on the jth capital asset purchased before time 0.
Total taxes (T(t)) paid by the firm, excluding those directly associated with debt finance, are as follows,
T(t) = t(t) {p(t) D [L(t),K(t),VjeJ] — w(t)
L(t) — p(t)
F[I(t),K(t)]} jeJ
—
, v(t)
(k(t)+z(t)) I(t) — A3(t,0)
(7)
jeJ
Net financing costs (F(t)) for the jth type of capital are
introduced into the model by augmenting the firm's cash flow to
9
reflect the acquisition and retirement of debt and net-of-tax interest payments.4 A firm that finances a proportion (b(s)) of its net-of-tax investment with debt will have its cash flow incremented by
X(s) =
b(s)
(l_k(s)-z(s)) vs(s) I(s) .
(8a)
Debt policy is exogenous to this model, and debt is retired at an exponential rate (TI). In period t, the cash flow devoted to retirements equals the amount of debt issued at time s (X(s))
multiplied by the "survival" factor (exp-rI (t-s)) and the retirement rate (TI), and summed from time t backward,
t f Ti X(s) exp(-i1(t-s)) ds .
(8b)
-00
payments at time t are the product of the amount of debt issued at time s surviving at time t and the interest rate (i(s)) prevailing at time s, summed from time t backward. In recognition Interest
of the deductibility of interest payments against income taxes, they are multiplied by one less the tax rate at time t, t
(1-t(t)) f i(s) X(s) exp(-TI (t-s)) ds . -00
This formulation of debt finance follows from joint work with Stephen King (1983).
(8c)
10 Equations
(8a-c) can be embedded in the firm's discounted cash
flow expression (2), and rearranged to obtain the following expression for the firm's net financing costs,
F(t) =
v(t) (l-k(t)—z(t)) (t) I(t) — B(t,O)
(9a)
,
ic J 00
v(t)
=
S
b(t){{ f exp(- f (p(u)+Tl) du) [(1-t(s)) i(t) +
t
11]
ds} - 1.O},
t
(9b)
0 B(t,O) = (1-t(t))
f i(s) X(s) exp(-'r (t-s)) ds -00
0
+
rf -00
(9c)
X(s) exp(-rl(t-s)) ds. t[O,oo)
Equation (9b) represents the potential subsidy for the purchase of investment goods provided by debt finance. If financial policy eliminates arbitrage opportunities between the net-of-tax cost of borrowing and the capitalization rate on debt, then ic(t) becomes unity, and the potential advantage of debt financing, arising from the tax deductability of interest payments, would be eliminated.
Equation (9c) represents the value of the net-of-tax interest and retirement payments at time t on debt acquired prior to time 0 and, when discounted by p(t) and integrated from zero to infinity, becomes the market value of the firm's debt at time 0 (B*(0)).
This
interpretation holds for any time paths of the variables, but becomes
11
apparent when we assume i(s), 'r(t), and p(t), are constant from time zero onward,5 B*(O) = [((l-t(O)) i(O) + 'ri) I (p(O) + TI)] FV(O) ,
(lOa)
0
FV(0) =
where
f X(s) exp(rs) ds ,
(lOb)
jeJ -
FV(O) is the face value of debt remaining at time 0.
If the
costs of debt and equity are equated through financial policy, then the term in braces in (lOa) is unity, and the face and market values of debt are identical. To complete our specification of the cash flow problem facing
the firm, we assume that movements in the capital stocks are governed by the following transition equations,
K(t) = I(t)
—
& K(t) .
VjcJ
(11)
The firm is assumed to maximize (2) constrained by (3), (4), (7), (9), and (11) and, in order to analyze the optimal choices of labor
and capital inputs, we construct the following current-value Hamiltoni an,
The general expression for the market value of debt is as follows, 00
B*(0) = FV(0)
0
f exp(- f (p(u)÷) du) { f co(s)[(l-t(t))i(s)+TI] ds} dt, 0
where
t 0
-00
o(s) is the percentage of the face value of debt issued in
period s.
12
t
H [L(t), I(t), K(t), .t(t), VjeJ] =
exp(-
f p(s) ds)
(1 2)
0
{(1-t(t)) {p(t)
[L(t),K(t),VjeJ} - w(t) L(t)
- p(t)
F[I(t),K(t)] } je J
v(t) I(t) + t(t) (I(t) -
—
K(t)) + A(t,O) — B(t,O)}
,
je J
v(t) = v(t) (l—k(t)-z(t)) (1+'qr(t))
te[0,oo)
where (12) is written in terms of the control (L(t), I(t)), state (K(t)),
and current-value co-state (p.(t)) variables, and (t) is the purchase price of new capital adjusted for the effects of taxes and financing.6 Note that A(t,O) and B(t,O) are predetermined in period t, and thus do not affect the firm's profit-maximizing decisions.
Necessary
conditions for the maximization of (12) are obtained by applying
6 In regard to debt finance, our formulation differs in two respects from that found in the Q models of Summers (1981) and Poterba and Summers (1983). First, our derivation permits the marginal and average levels of debt finance to vary, and thus average Q (or 2 in the current notation) will reflect the time variation in the market value of debt. Second, the specification of the purchase price of new capital allows for the possibility that financial policy may not eliminate the subsidy associated with debt finance. When the percentage of debt finance is constant (as has been assumed in previous studies), it is unlikely that this subsidy will be zero, especially when nominal interest payments are tax deductible.
13
Pontryagin's Maximum Principle and, for purposes of the present analysis, we consider the following conditions pertaining to labor and the jth type of capital,
L[t] =
w(t) I p(t) ,
00
=
ft
(13a)
5
exp(- f (p(u)+) t
du) (1-t(t)) p(t)) {cIKJ[s] - FKJ[s]} ds
(13b) +
(l-t(t)) p(t)) F1[tJ .
(13c)
VjcJ
(13a) is the familiar marginal productivity condition for a variable factor of production. The marginal benefit of an additional Equation
unit of capital is defined in (13b) as the sum of current and future marginal products weighted by the rates of discount and Along the optimal path, this marginal benefit must equal the total marginal costs of acquiring capital. These involve the sum of purchase and marginal adjustment costs, and the requisite depreciation.
equality is given by (13c).
14
III. THE MULTI-CAPITAL 0 MODEL - SPECIFICATION
The optimizing conditions associated with the maximization of (12) form the basis of the Multi-Capital Q model of investment. dividing both sides of (13c) by v(t). we derive the positive
By
relationship between investment, as embedded in the adjustment cost function, and marginal Q, the ratio of the shadow price for the jth capital good to its net purchase price.7 Since this equation contains an unobservable variable, it is of little immediate use in applied work. For a firm that uses only equity finance and one
capital input, Hayashi (1982a) has developed the conditions under which the unobserved shadow price of capital can be related to financial market data.
These conditions include that the production
and adjustment cost technologies exhibit constant returns to scale,
the firm participates in perfectly competitive output and factor markets, capital depreciates exponentially, and the discounted values of the capital stocks are constrained by transversality conditions.8 Modifying Hayashi's proof to incorporate multiple capital goods and debt finance, we obtain the following relationship between shadow
prices and financial data,
This relationship has been noted by, among others, Abel (1979), Mussa (1977), Sargent (1979), and Yoshikawa (1980). 8 The transversality condition is written as follows, T
Lim exp(- f p(s) ds) j.i(T) K(T) = T—*oo
0
0
.
VjcJ
15
V(O) =
(O) K(O) + A*(O)
-
B*(O)
(14a)
,
ic J
00
A*(O) =
f o
exp(- f p(s) ds) 0
A(t,O) dt,
(14b)
B(t,O) dt,
(14c)
jcJ
00
B*(0) =
f
exp(- f p(s)
o
0
ds)
jeJ
The intuition behind this results is rather straightforward.
The
assumption of competitive markets ensures that, after period 0, the firm is unable to earn any profits, and the equity value of the firm is determined by the quasi-rents from the stocks available at the beginning of the planning period. The most important of these is fixed capital and, under constant returns to scale, the marginal and average return to capital are identical. Thus, the value of the jth capital good is the product of its shadow price and the existing stock.
Assuming momentarily that the firm uses only one capital good and that A*(0) and B*(0) are zero, we see that (14a) delivers the basic relationship between marginal and average Q, the latter defined as the ratio of the financial value of the firm to its replacement cost, average Q
V(0)
/ (0) K(0) =
j.t(0)
/ (O)
marginal Q.
(l5)
16
More generally, the equity value depends on the other capital goods and their shadow prices, plus the discounted value of tax depreciation allowances due to past investments (A*(O)), less the discounted liability due to past financing committments (B*(O)). To obtain an investment equation amenable to testing, we need
to impose some assumptions about the intertemporal technology, and choose the following parameterization of the adjustment cost function s,
F3[I(t),K(t)J = (yf2) [I(t)/K(t) — J}2 K(t)
VjeJ
(16)
noted in the introduction, past studies have found that lagged variables have been significant in Q investment equations, and As
lagged dependent variables are included in the general econometric specification to represent any dynamics not fully captured by (16). \Vhile these lagged variables compromise the structural
interpretation of the investment equations, they are included in the general specification because their statistical significance serves as a test for misspecification in the expanded model. Furthermore, it would appear preferable to account explicitly for the dynamics rather than doing so implicitly through a GLS correction. Combining (13c), (14), and (16), we obtain the following system of investment equations for the J capital goods,
17
I(t) =
K + (1 /y3) f (t) -
(Yk'Y) lk(t)
(17a) + (Yk• Sk/Yj) Kk(t) +
(t) =
I(t-1) + c(t) ,
{V(t) + B*(t) - A*(t) jeJ
Vj,kei,
v(t) K(t)} / ((1-t(t))
p(t),
(17b)
where the k subscript represents all but the jth capital good and (17) corresponds to the case where J=2. A white-noise error term (c(t))
reflects non-systematic variations in I(t) and approximation errors that have arisen in the development of the model,9 and is the parameter for the lagged dependent variable. Equations (17) highlight that, in the Multi-Capital Q model, market value data will not prove to be a "sufficient statistic" for
signaling investment expenditures for any particular capital input. In (17b), (t) is defined as the difference between the financial value
of the firm (adjusted for tax depreciation) and the replacement cost of the capital stocks all deflated by the net-of-tax output price. (Note that the same 1 (t) enters all J investment equations and, if adjustment costs were valued by the price of investment goods An additive error term can be derived from the theoretical model if we assume that technology shocks affect either the production function (Hayashi, 1982b) or the adjustment cost function (Poterba and Summers, 1983). The maximization problem has not been formulated with these additional assumptions because neither result in any restrictions on the estimation procedure, other than to warn of the omnipresent possibility of correlation between error terms and endogenous regressors (i.e., Ik(t)).
18
rather than output, (cf., (15)).)
(t) would be replaced by average Q less unity In (17a), a given value of (t) indicates the profitable
level of investment activity for the entire firm.
The own capital
stock influences gross investment positively due to replacement needs. The regressors subscripted by k affect I(t) through adjustment costs. The higher Ik(t), the greater the adjustment costs for the kth capital good and, hence, the fewer resources available for investment in the jth capital good. Since a larger capital stock lowers adjustment costs, Kk(t) has a positive effect on investment
The failure to include these latter variables in econometric equations implies that conventional formulations of the expenditures.
Q model may be misspecified.
The stronger the association between
(t) and investment expenditures for the kth capital good, the greater the upward bias in Yj.10 The econometric results presented in the next section will allow us to assess the extent of this bias and the degree to which the Multi-Capital model attenuates the other empirical problems associated with Q theory.
10 Alternatively, it has been noted that the large estimated values of are an inevitable outcome of relating volatile financial market data to the less variable investment series (Shapiro, 1986, p. 531). Insofar as the flow variables are strongly, positively correlated with 1 (t), the estimated 7j's from (17), ceteris paribus, should fall.
19
IV. ECONOMETRIC RESULTS
The Multi-Capital Q
model
was estimated with data for
nonfinancial corporations over the period 1950-1978 for three capital goods - equipment, structures, and inventories. The length of the sample and the number of capital goods were determined by data availability, and detailed information concerning data sources is For each capital input, (17a) was scaled by K(t) to eliminate the possible spurious correlation arising from the provided in the Glossary.
trends in the flow and stock variables (Granger and Newbold, 1974).
Since no estimation technique dominates under even modest violations of maintained assumptions, results are reported for a variety of estimators.
Ordinary Least Squares estimates of the Conventional (CV) and Multi-Capital (MC) models are presented in Table II, columns 1-3 and 4-6, respectively.
For the CV model, the bulk of the explanatory
power of the Q model can be traced to the structures equation. As assessed by the rn-statistic,11 the residuals are serially correlated and, contrary to a strict interpretation of Q theory, the lagged dependent variables in the equipment and structures equations are significant. The results for the MC model are mixed. The 2's are improved substantially, and serially correlated residuals are absent. rn-statistic is distributed t under the null hypothesis of no The rn-statistic is calculated in a regression of the residuals from an equation with a lagged dependent variable (e.g., (17a)) on all of the explanatory variables in the initial regression plus the lagged residual; m is the t-statistic on the lagged residual. Performing a t-test on this coefficient is asymptotically equivalent to the Durbin h-test. However, it can be calculated for all possible values of the estimated parameters, and has performed better than the h-statistic in Monte Carlo experiments (Harvey, 1981, p. 276). 11
The
serial correlation.
20
However, the only significant coefficient on c(t) appears in the structures equation, which also contains a significant lagged dependent variable. While a number of the additional flow and stock variables are significant, most are of the wrong sign.
The CV
specification is nested within the MC and, for each of the three
investment equations, the zero restrictions on the four additional regressors (Ik(t), Kk(t), Vk,jEJ,
are rejected at the 1% level.
Key to our development of the Multi-Capital Q model is the
assumption that the adjustment cost parameters differ among the capital inputs, and two tests suggest that the null hypothesis --
= Yk' Vj,kEJ,
is
rejected by the data.
First, the rejections o
the CV model reported above are strong evidence in favor of the MC
specification.'2 Second, we can interpret the aggregate equation (1) as a restricted case of the separate MC equations (17) and, regardless of which capital good served as the dependent variable, the restricted model was always rejected at the 1% level.13 Estimates for the MC model may be biased by correlations
between error terms and the investment flows (the capital stocks 12 These results hold whether or not a lagged dependent variable is included as a regressor. 13 To ensure that (1) is nested within (17a), the separate MC equations have all been scaled by the aggregate capital stock instead of K(t) and the constant term in the restricted model is interpreted as a weighted average of the depreciation rates for the separate capital goods. The tests were conducted for each of the capital goods four different ways: with or without a lagged dependent variable, and for two (equipment and structures) or three (equipment, structures, and inventory) capital goods. This aggregate Q investment equation stands in contrast to Wildasin's (1984) Proposition II, which is based on an aggregate investment equation specified as the ratio of nominal variables.
21
and (t), dated at the beginning of the period, are predetermined), and Table III contains two sets of estimates that avoid this potential Reduced form estimates are contained in the first three columns, and all of the coefficients on (t) are significant. An problem.
implication of the adjustment cost technology is that these coefficients should be lowest for structures (representing the largest
adjustment costs) and highest for inventories, a prediction borne-out by the estimates. Instrumental variables (IV) regressions for the MC model are presented in columns 4-6 of Table III and, relative to OLS, these estimates remain essentially unchanged.14 By recognizing the contemporaneous correlation between the cs we can enhance the efficiency of the coefficient estimates, and can conduct an additional test of the specification.
Three-Stage Least
For the MC model, all of the coefficients on Q(t) are significant at the 1% level, Squares (3SLS) estimates are presented in Table IV.
though the coefficient in the equipment equation is negative. Coefficients on the lagged dependent variables and tests for serially correlated residuals are both insignificant.
The system of equations characterizing the MC model has 12 regressors not contained in the
CV model, and 11 of these are statistically significant, 4 of which are of the correct sign. Not surprisingly, a comparison of the CV and MC models based on the differences in their Criteria, distributed X2(12),
14 The instrument list included all of the predetermined variables in the three equation system plus keq(t), zeq(t), t(t). The R2's for the first-stage regressions were no lower than .78. When the instrument list was reduced to keq(t), zeq(t), t(t), and all predetermined variables in a given equation, the parameter estimates were largely unaffected.
22
rejects the restrictions defining the Conventional Q level.
model
at the 1%
Further support for the MC model is obtained by comparing
the IV and 3SLS estimates with the test proposed by Hausman (1978), Durbin (1954), and Wu (1973). If misspecification is present in the system, the 3SLS estimates will differ substantially from the corresponding IV estimates. For the three equation system, the HDW
test statistic is distributed 2(21), and equals 6.372 (P >
.999),
indicating that the null hypothesis of no misspecification can not be rejected. 15 To gain additional insight into the MC model, we examine the
role of inventory capital, which has been excluded in a number of previous Q studies.
Table V contains estimates of equation systems
for equipment and structures, and the results change little for the CV model (columns 1-2) or for the MC model when the inventory terms remain as regressors (columns 3-4). These inventory terms are
removed in the regressions reported in columns 5-6, and the coefficients on the lagged dependent variables increase sharply.
Since inventories play an important role in buffering the firm against shocks, failing to include these terms may lead to misspecified dynamics captured in an ad hoc manner by lagged variables in conventional Q models. Formally, a comparison of the Criteria for the This statistic is based on the IV estimates in Table III, columns 4-6, and the 3SLS estimates in Table IV, columns 4-6. The test statistic was computed with the 3SLS residual covariance matrix, and can be interpreted as the Lagrange multiplier version of the HDW test. In comparing the IV and 3SLS estimates, note that the standard errors of the estimated coefficients from the single equation 15
estimators (Tables II and III) are adjusted by the degrees of freedom, whereas the the estimated coefficients from the systems estimators (Tables IV and V) are adjusted by the sample size.
23
models in Table V rejects the hypothesis that inventories can be excluded.
Complementary evidence is provided by the HDW test
statistics of 8.261 (P > .875) for columns 3-4 and 13.721 (P > .186) for columns 5-6. While both tests are below conventional significance
levels, the substantial increase in the latter P value confirms the importance of the inventory variables.
24
V. SIMULATIONS
To evaluate further the Multi-Capital Q model, we examine in
this section the response of investment expenditures to changes in V(t) (hence (t)) in both the CV and MC models. We assume that V(t) increases unexpectedly by $1, perhaps due to a lump sum tax rebate,16 and use the estimated parameters to calculate the increments to investment and the capital stocks.17
These calculations
will be only approximate because we are not constraining the
solution to be consistent with the eventual steady-state (i.e., remain on the stable manifold). A comparison of our method to that used by Summers (1981, Table 6) indicates that the understatement of the accumulated capital stock is small: the bias is approximately 0% during the first five years, 1% in the tenth year, and 13% in the fiftieth year.
The simulation results are reported in the Tables at the bottom three rows for each equation, and the response of investment is quite slow. Ten years after the $1 shock, increments to the accumulated sum of the three capital stocks range between $.19 and $.37.
These
results do not vary systematically with the CV or MC specifications,
and are quite consistent with the value of $.32 from the equation 16 These simulation results are comparable to and consistent with the evidence from other econometric investment models discussed in Chirinko (1986).
17 Since the lagged dependent variables are not part of the formal Q model and lead to explosive behavior, they have been excluded in the simulations. The system of investment equations (17) was augmented by the capital accumulation equations (11) with geometric depreciation rates of .13, .04, and 0.0 for equipment, structures, and inventory, respectively. When the three capital inputs are aggregated, the depreciation rate was set to .061.
25
based on aggregate capital (1). A result not apparent in the aggregate equation is that inventories are accumulated at a disproportionately fast rate in the conventional model. For the sample period, the inventory stock is 25% of the aggregate, but the simulated changes in inventories are between 51-56% (CV models) and 19-32% (MC models). In sum, one finds that the modifications to
the Q model undertaken in this paper do little to attenuate the unreasonably sluggish response of investment to variations in L (t).
26 VI. SUMMARY AND CONCLUSIONS
The poor empirical performance of Q models has led us to reexamine the conventional framework. By recognizing the
possibility that the value of the firm depends on two or more capital inputs whose adjustment cost technologies may differ, we have derived an econometric specification that nests the Conventional
model as a special case.
Econometric evidence was generated with
equipment, structures, and inventories considered as capital inputs, and the Conventional model was rejected in favor of the MultiCapital specification. However, for a number of the variants of the Multi-Capital model, lagged variables proved significant, serial correlation in the residuals was important, and a number of the regressors had incorrect signs.
Furthermore, as has been the case in
previous work with Q models, the response of the capital stock to unexpected changes in the economic environment remained unreasonably slow.
The results reported in this paper thus provide substantial support for the Multi-Capital model as a useful extension to the Q In particular, framework, but indicate that further work remains. inventories were shown to play a key role in the modeling of fixed investment expenditures within the Q framework. One possible direction would be to modify the technology in order to recognize a buffer stock role for inventory investment but retain a traditional
27 adjustment
cost approach for equipment and structures.18
Alternatively, the dynamics of fixed capital accumulation may be too
complex to be captured adequately by the adjustment cost technology. A more satisfactory model of capital accumulation may thus have to be based on an intertemporal technology that recognizes explicitly delivery, expenditure, and gestation lags and the possibility of non-exponential capital decay.
the availability of suitable data, the MC framework could be extended to other capital inputs such as research and development, human capital, or advertising/goodwill. However, in the event that the data associated with these types of capital are plagued by substantial measurement error, their inclusion may compromise the other coefficient estimates (cf., Fisher, 1980, p. 158). 18
Given
R— 1
REFERENCES
Abel, Andrew B., Investment and the Value of Capital (New York: Garland Publishing, 1979). and Blanchard, Olivier, "The Present Value of Profits and Cyclical Movements in Investment," Harvard University (October 1984). Chirinko, Robert S., "On the Inappropriateness of Empirical Q Investment Equations," Northwestern University (June 1982a). "The Not-So-Conventional Wisdom Concerning Taxes, Inflation, and Capital Formation," National Tax Association - Tax
Institute of America Proceedings _____
1982b,
272-281.
"Will 'The' Neoclassical Model of Investment Please Rise?:
The General Structure of Investment Models and Their Implications for Tax Policy," in Jack Mintz and Douglas Purvis (eds.) The Impact of Taxation on Business Investment (Kingston, Ontario: John Deutsch Institute of Economic Policy, 1986). King, Stephen R., "Hidden Stimuli to Capital Formation Debt and the Incomplete Adjustment of Financial Returns," Cornell Ufliversity Working Paper No. 301 (May 1983).
________
and
Ciccolo, John H., "Four Essays on Monetary Policy," Unpublished Ph.D. Dissertation, Yale University (May 1975).
Clark, Peter K., "Investment in the 1970's: Theory, Performance, and Prediction," okings Papers on Economic Activity (1979:1), 73-124. Durbin, James, "Errors in Variables," Review of the International
Statistical 1nstitut 22 (1954), 23-32. Fischer, Stanley, "A Note on Investment and Lagged Q," Massachusetts Institute of Technology (December 1983).
R— 2
Fisher, Franklin M., "The Effect of Simple Specification Error on the Coefficients on 'Unaffected' Variables," in L.R. Klein, M. Nerlove, and S.C. Tsiang (eds.) Ouantitative Economics and Development: Essays in Memory of Ta-Chung Liu (New York: Academic Press, 1980), 157-163. Granger, Clive W.J., and Newbold, P., "Spurious Regressions in Econometrics," Journal of Econometrics 2 (July 1974), 111-120.
Hall, Robert E., "Investment, Interest Rates, and the Effects of Stabilization Policies," Brookings Papers on Economic Activity 1977:1), 61-103. Hansen, Lars P., and Sargent, Thomas J., "Formulating and Estimating Dynamic Linear Rational Expectations Models," Journal of Economic Dynamics and Control 2 (February 1980), 9-46. and Singleton, Kenneth J., "Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models," Econometricp 50 (September 1982), 1269-1286. Harvey, Andrew C., The Econometric Analysis of Time Series (New York: John Wiley, 1981). Hausman, Jerry A., "Specification Tests in Econometrics," Ecpnometricp 46 (November 1978), 1251-1271.
Hayashi, Fumio, "Tobin's Marginal q and Average q," Econometrica 50 (January 1982a), 213-224. —, "Real Wages, Employment, and Investment in the Theory of
Adjustment Costs," Northwestern University (April 1982b). Jorgenson, Dale W., "Capital Theory and Investment Behavior," American Economic Review 53 (May 1963), 247-259.
Keynes, John Maynard, The General Theory of Employment. Interest and Money (New York: Harcourt, Brace, 1964). Lucas, Robert E., Jr., "Econometric Policy Evaluation: A Critique," IJi Phillips Curve and Labor Markets, Carnegie-Rochester Conferences in Public Policy 1 (1976), 19-46; reprinted in Studies in Business-Cycle Theory (Cambridge: MIT Press, 1981).
R—3
Mussa, Michael, "External and Internal Adjustment Costs and the Theory of Aggregate and Firm Investment," Econornica 44 (May 1977), 153-178. Nerlove, Marc, "Lags in Economic Behavior," Econometrica 40 (March
1972), 221-251. Poterba, James M., and Summers, Lawrence H., "Dividend Taxes, Corporate Investment, and Q," Journal of Public Economics 22 (March 1983), 135-167.
Sargent, Thomas J., Macroeconomic Theory (New York: Academic Press, 1979). Shapiro, Matthew D., "The Dynamic Demand for Labor and Capital," Quarterly Journal of Economics 101 (August 1986), 513-542.
Summers, Lawrence H., "Taxation and Corporate Investment: A qTheory Approach," Brookings Papers on Economic Activity (1981:1), 67-127. Tobin, James, "A General Equilibrium Approach to Monetary Theory," Journal of Money. Credit. and Banking 1 (February 1969), 1529.
\Vildasin, David E., "The q Theory of Investment with Many Capital Goods," American Economic Review 74 (March 1984), 203-210.
Wu, D., "Alternative Tests of Independence Between Stochastic Regressors and Disturbances," Econometrica 41 (1973), 733750. Yoshikawa, Hiroshi, "On the 'q' Theory of Investment," American Economic Review 70 (September 1980), 739-743.
G— 1
GLOSSARY SOURCES:
BAL -
of Governors of the Federal Reserve System, "Balance Sheets for the U.S. Economy, 1945-82," (October 1983). Board
BEAU -
Unpublished
data provided by the Bureau of
Economic Analysis.
BS -
U.S. Department of Commerce, Bureau of Economic
Analysis, Business Statistics (1982, 1971).
CEA -
of Economic Advisers, Economic Report of the President (Washington: U.S. Government Printing Office, 1984).
CS -
Council
Corcoran,
Patrick J., and Sahling, Leonard, "Business Tax
Policy in the United States: 1955-1980," Federal Reserve Bank of New York, Research Paper No. 8102 (September 1981), and unpublished data provided by the authors. For a related publication, see Corcoran, Patrick J., "Inflation, Taxes, and the Composition of Business Investment," Federal Reserve Bank of New York Quarterly Review 4 (Autumn 1979), 13-24.
NIPA -
U.S.
Department of Commerce, Bureau of Economic
Analysis: (1976-1978), Survey of Current Business 62 (July 1982); (1940-1975), The National Income and Product Accounts of the United States 1929-1976 Statistical Tables (September 1981). John J., "Marginal Federal Personal and Corporate Income Tax Rates in the U.S., 1909-1975," Journal of ST
-
Seater,
Monetary Economics 10 (November 1982), 361-381.
S&P -
Standard
& Poor's Statistical Service, Security Price
Index Record, 1984.
VF -
von Furstenberg, George M., "Corporate Investment:
Does Market Valuation Matter in the Aggregate?," on Economic Activity (1977:2), 347-397.
Brookirigs Papers
C— 2
LEGEND:
BOP -
beginning column.
of the period.
Cflow over the period. L - line. MOP - middle of the period. T - table. FLO -
DEFIN ITIONS: -
A*
Present
discounted value of tax depreciation
allowances from assets purchased prior to period t, BOP: Cs. b
- Leverage -
ratio: B* / (B* + V).
dollar market value of debt issued by nonfinancial corporate business, BOP: (NETINT * (MIP / INT)) / i). B*
Current
-
DIV
dollar dividends for nonfinancial corporate business, BOP: NIPA, T1.13, L31. Current
-
DSPC
Standard
& Poor's dividend-common stock price
ratio, BOP: S&P, p. 127.
DSP -
Standard
& Poor's dividend-preferred stock price
ratio, BOP: S&P, p. 118. -
nominal interest rate on corporate Aaa bonds, BOP: NIPA, S-16, and BS. i
Moody's
-
dollar investment for nonfinancial corporate business (j = equipment, structures, inventories), FLO: BEAU. Ii
Constant
-
dollar net interest paid by nonfinancial corporate business, FLO for the previous period: NIPA, T1.13, L35. INT
Current
C— 3
k-
Rate
of investment credit (j =
equipment, structures):
CS, p. 54. -
K
dollar replacement value of the capital stock for nonfinancial corporate business (j equipment, structures, inventories), BOP: BEAU. Constant
K$ -
dollar replacement value of the capital stock
Current
for nonfinancial corporate business (j =
equipment, structures,
inventories), BOP: BAL, T705, L4. -
MIP
Current
dollar net monetary interest paid by nonfinancial corporate business, FLO for the previous period: NIPA, T8.7, L7 less L25.
NETINT -
dollar net interest paid by nonfinancial corporate business, BOP: NIPA, Tl.13, L35. -
NFA
Current
dollar noninterest bearing net financial assets of nonfinancial corporate business, BOP: BAL T705, L9, L16, L17, L18, less L37, L38, L39. -
Current
Rate
of debt retirement: .1087, which implies that 90%
of a debt issue would be retired after 20 years. p
-
p
-
Implicit price deflator for gross domestic purchases, BOP: NIPA, T7.3, L15. Firms' nominal rate of discount, BOP: (wdjv DSPC + DSP) + LAG4(SP/SP), where the latter variable equals
the (l-wdjv) mean of the percentage change in SP over the previous four periods.
SP -
Standard
& Poor's composite stock price index, MOP:
CEA, TB-90.
t
Rate
-
Tax
of federal taxation of corporate income: ST, T2, C6.
credits and deductions on capital services = equipment, structures): (1 - kJ - t z).
rj
U
-
G-4
v-
Price
V-
Current
index for investment expenditures on asset j: calculated implicitly with K$.
dollar market value of equity for nonfinancial corporate business, BOP: DIV / (wdjv DSPC + (l-wd1) DSP). -
Difference
between the value of the firm evaluated on
financial markets and the net-of-tax replacement value of its assets, BOP: [(V + B*) -
Wdjv
(tK$ +
Percentage
+ (1t)K1$ + NFA)] / (l.t) p.
of dividends paid on common stock: VF,
p. 358, fn. 11, extended for the current study. -
for the purchase of investment goods provided by debt finance, BOP: b{[((1-t)i + ri) / (p + 1)] - 1.O}. -
Subsidy
discounted value of current and future tax depreciation allowances per dollar of investment in period t Zj
(j =
Present
equipment, structures): CS, Appendix E.
TABLE I SERIAL CORRELATION COEFFICIENTS
EQUIPMENT. STRUCTURES. INVENTORIES*
Lag Length
Equipment
Structures
Inventories
(1)
(2)
(3)
1
.728
.896
.263
2
.425
.753
-.200
3
.374
.658
-.189
* Sample period 1953-1978.
All series scaled by their own capital stocks. Critical values are .496 and .388 at the 1% and 5% levels, respectively.
TABLE II
CONVENTIONAL AND MULTI-CAPITAL Q MODELS
ORDINARY LEAST SQUARES ESTIMATES
Corw1bn Variable or Sta1ist
EQ (1)
.0041 (.0020)
ST (2)
Multi-CtaI
IN (3)
•0029a
.0051
(.0010)
(.0035)
- — ——
Keq(t)
EQ
ST
IN
(4)
(5)
(6)
-.0012 (.0018)
0025a (.0008)
.0031 (.0027)
— •2706a — (.1000)
13900a (.1744)
.0449
.0692
.0795a
-.0304
(.0312)
(.0545)
(.0266)
(.1064)
st(t)
— —
(.3114) — (.4812)
K5t(t)
.0254
.1567a
(.0128)
(.0452)
.6410 — -.8944
—— —
l(t) Kin(t)
(t-1)
2485a
(.0631)
-.0913 —
(.0671)
(.0680) —
—
0372a
2625a
-.01 40
——
.5282a
(.0105)
(.0978)
(.0556)
(.1609)
—
.0868
7820a (.1818)
-———
——-
— —
(.1382)
(.1832)
—
•7398a
lst(t.l)
Iin(tl)
5096a
0942a
(.0197)
-———
——
.1558
—
(.1941)
——
——-
(.1451) .1490 (.1419)
.4324
.7154
.0331
.8259
.8527
.7275
1.8572
2.7819
3.9303
1.2659
.2230
-.2731
.5878
.0793
3.3425
.1526
.0347
.7970
82.155
111.20
56.954
101.71
123.18
77.741
secondyear fifthyear
.0115
.0082
.0144
.0089
.0082
.0137
.0395
.0694
.0593 .1414
.0327 .0634
.0329 .0724
.0341
tenth year
.0311 .0649
m Res.
Sum. Sq.
Criterion
Simulations (dKLi .0550
*Estimates based on equation (17). The dependent variable is indicated by the column heading. All
explanatory variables are scaled by the capital stock for the dependent variable; thus, for a given equation, the corresponding capital stock represents the constant term. Sample period 1950-1978. Standard errors in parentheses. Residual Sum of Squares multiplied by i02. a Significant at the 1% level. b Significant at the 5% level.
TABLE III
MULTI-CAPITAL Q MODELS
REDUCED FORM AND INSTRUMENTAL VARIABLES ESTIMATES
Reduced Form
Instrumental Variables
Variable or Staustc
(t) I(t) Keq(t)
EQ
ST
(1)
(2)
0082a (.0021)
K8t(t)
l(t)
K(t) Iq(t1) Ist(tl)
lin(tl)
0037a
.01 19a
(.0009)
(.0034)
—
—
——— -.1631
.lO4Ba
-.1936
—— — ——
(.0946) lst(t)
(3)
(.0403)
(.1526)
EQ
ST
IN
(4)
(5)
(6)
-.0022 (.0026)
0029a (.0009)
.0047 (.0033)
3226b
— (.1449) .0849
.0864a
(.0719)
(.0281)
13813a (.1975)
(.1188)
.7910 — -1 .2534 (.4994) — (.6126)
2023a
oo
.472
1832a
•1083a
(.0697)
(.0297)
(.1118)
(.0716)
(.0242)
(.0752)
— ———
(.0902)
(.1005) —
.1128
.0718
-.4170
•2767a
-.0247
.4665a
(.1577)
(.0653)
(.2574)
(.1020)
(.0692)
(.1812)
12828a
.1408
17194a
.0019
(.3308)
(.1388)
(.5285)
(.2230)
17448a
.2318
5863a
2g245a
-.1731 —
3532b ——-
(.4894)
(,2048)
(.7812)
-.0655
-.0225
.0620
———-
——
.1491
———-
——-
(.1444)
(.1573)
(.1406)
(.0604)
(.2261)
.5911
.7968
.3753
m
.6759
2.4944
-.3128
.3913
-.3978
-.3189
Res. Sum. Sq.
.3583
.0479
1.8274
.1628
.0381
.8199
89.334
118.51
65.709
10.334
8.9431
8.5567
.0231
.0104 .0407 .0779
.0335 .0704 .0884
.0108 .0459
.0086 .0369 .0825
.0173
Criterion
———
Simulations (dKi secondyear fifth year tenth year
.0796 .1315
.1105
.0482 .0902
Estimates based on equation (17). The dependent variable is indicated by the column heading. All explanatory variables are scaled by the capital stock for the dependent variable; thus, for a given equation, the corresponding capital stock represents the constant term. Sample period 1950-1978. Standard errors in parentheses. Residual Sum of Squares multiplied by i02. Criterion for IV divided by the variance of the regression error. a Significant at the 1% level. b Significant at the 5% level.
TABLE IV CONVENTIONAL AND MULTI-CAPITAL 0 MODELS THREE STAGE LEAST SQUARES ESTIMATES
MCtaJ
Converitbnal
Variable or Statstc
(t)
I(t) Keq(t)
1tt)
EQ (1)
0041b (.0019)
ST (2)
0023b (.0009)
IN (3)
.0053 (.0033)
— —— ——
Criterion
(.0018)
(.0007)
(.0026)
5220a
— (.1060)
fifth year tenth year
(.1235)
(.0552)
(.0230)
(.0859)
— —— —
1•2157a — 1g877a (.3300) — (.4645) 2343a
.0086
— — ——
— — 8191a -———
•1427a
•3787a
(.0423)
(.0168)
(.0568)
6224a
.3138a
(.0533)
(.0700) —
0391a
2155a
-.0841
(.0087)
(.0752)
(.0547)
—
-.0066
——-
(.0789)
— •9223a
——--
.2089
— (.1 140)
———
(.1118)
——-
————
.1064
— —-—-
(.1321)
.0007
— (.1400)
———-
—-
(.0491)
1.7795
2.0914
3.8745
.0121
-.6636
-.3819
.5888
.0846
3.3511
.1961
.0525
1.0189
——-—-—-——- {61.364} ——-————----—-
(21 .560}
Simulations (dK.{ secondyear
1 5876a
(.0209)
I(t-1)
Res.Sum.Sq.
.007
g8a
(.1213)
m
•0032a
.0g8ga
I,(t)
151(tl)
.0042b
IN (6)
(5)
•1405b
(.0105)
I(t-1)
ST
.0385
K5t(t)
K1(t)
EQ (4)
.0115 .0392 .0680
.0065 .0241
.0483
.0149 .0618 .1486
.0212 .0856
.0088 .0659
.l355c
.l624c
.0357 .0557
•Estimates based on equation (17). The dependent variable is indicated by the column heading. All explanatory variables are scaled by the capital stock for the dependent variable; thus, for a given equation, the corresponding capital stock represents the constant term. Sample period 1950-1978. Standard errors in
parentheses. Residual Sum of Squares multiplied by i02. a Significant at the 1% level. b Significant at the 5% level. C The insignificant coefficient set to zero.
TABLE V CONVENTIONAL AND MULTI-CAPITAL 0 MODELS
ThREE STAGE LEAST SQUARES ESTIMATES*
Conventbnai
Variable or Statisti
ST
EQ
(1)
(2)
(3)
(.0019)
0023b (.0009)
.0332
l5t(t)
— .0085
— (.0106)
l(t)
K(t) 8503a
(.1514)
Ist(t1)
EQ
ST
(5)
(6)
OO3
.008
.0024a
(.0008)
(.0036)
(.0007)
— (.1086) (.0260)
I(t-1)
0046b (.0020)
ST
(4)
— 5087a
leq(t)
Kst(t)
Multi-Capital
________________________________________
EQ
•0041b
Keq(t)
-
— .105c — (.0455)
1481b
•0g35a
-.0484
.0473a
(.0571)
(.0239)
(.0846)
(.0123)
1.3206a
-1.0474 —
(.3713)
(.7165)
2544a
•1356a
.1395
(.0530)
(.0186)
(.1020)
6231a
2g28a
(.0760)
(.0774)
2472a
-.0857
(.0860)
(.0557)
-.1159
10654a
(.1724)
(.2830)
0658a
(.0138)
— 5876a
— •9236a
— (.1033)
(.1151)
———
(.1273)
1.6505
2.2302
.4390
-.6971
1.4557
1.0480
.5910
.0847
.2022
.0489
.7537
.0421
l(t1)
m
Res. Sum. Sq.
Criterion
——— (40.631)
——- (20.723) ———
——— {35.883}
-.0055 -.0278 -.0856
.0152 .0518 .0922
Simulations (dK1fdc1
second year fifth year tenth year
.0115 .0393 .0690
.0065 .0244 .0498
.0057 .0239 .0618
.0083 .0303 .0600
Estimates based on equation (17). The dependent variable is indicated by the column heading. All explanatory variables are scaled by the capital stock for the dependent variable; thus, for a given equation, the corresponding capital stock represents the constant term. Sample period 1950-1978. Standard errors in
parentheses. Residual Sum of Squares multiplied by i02. a Significant at the 1% level. b Significant at the 5% level.
